Neural Network Model for the Prediction of Water Aquifer

88

Energy & Fuels 1999, 13, 88-98

Neural Network Model for the Prediction of Water Aquifer Dimensionless Variables for Edge- and Bottom-Water Drive Reservoirs Ibrahim S. Nashawi* and Ali Elkamel College of Engineering and Petroleum, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Received May 27, 1998. Revised Manuscript Received October 14, 1998

Accurate estimation of water influx into a petroleum reservoir is very important in many reservoir-engineering applications, such as material balance calculations, design of pressure maintenance programs, and advanced reservoir simulation studies. These applications have relied heavily on the classical work of van Everdingen and Hurst for edge-water drive reservoirs and on the results presented by Coats and Allard and Chen for bottom-water drive reservoirs. However, for both types of reservoirs, the determination of the values of water influx is not a straightforward task. Table lookup and interpolation between time entries are needed, and furthermore, for finite aquifers, interpolation between tables may also be required. This paper presents neural network (NN) models for the prediction of dimensionless water influx and dimensionless pressure for finite and infinite edge- and bottom-water drive reservoirs. Several neural network architectures using back-propagation with momentum for error minimization were investigated to obtain the most accurate results. In order for these NN models to be applied for a wide range of systems, dimensionless groups characterizing water influx were employed. The advantage of the proposed NN models is providing accurate results in minimum time. Furthermore, they can be easily integrated within general reservoir management programs to determine the aquifer effect on oil and gas production.

1. Introduction Petroleum reservoirs are often surrounded from the edge or the bottom by water aquifers that support the reservoir pressure through water influx. In response to a pressure drop in the petroleum reservoir, the water aquifer reacts to offset, or retard, pressure decline by providing a source of water influx or encroachment. To determine the effect that an aquifer has on the oil and gas production, it is important to estimate the amount of water that has entered into the reservoir from the aquifer. Such calculation is not a simple and risk-free task due to the involvement of many unknown parameters. For instance, aquifer pressure, thickness, permeability, porosity, shape, and areal extent are usually all unknown variables. Furthermore, there are two different aquifer models that are classified according to the flow geometry as either edge-water or bottom-water drive. These models have completely different flow behavior. Consequently, it is very important to identify the appropriate model in order to obtain accurate results. Therefore, the type of the water aquifer, its size, its properties, and the amount of water that it can deliver into the reservoir for a certain pressure drop during a specific period of time affect the entire production life of the reservoir. A good knowledge of the aquifer properties, specifically the amount of water that it can provide into the reservoir, dictates the production schedule and the development strategies that need to be implemented in order to optimize oil recovery. Many authors1-13 have presented different models for estimating the water influx. These models apply to

different flow regimes, including steady-state,1 modified steady-state,2 pseudo-steady-state,3 and unsteadystate.4-5 van Everdingen and Hurst4 presented the most commonly used water-influx model. This model is basically a solution of the radial diffusivity equation; thus, it yields an accurate estimate of water enchroachment for (1) Schilthius, R. J. Active Oil and Reservoir Energy. Trans. AIME 1963, 118, 33-52. (2) Hurst, W. Water Influx Into a Reservoir and Its Application to the Equation of Volumetric Balance. Trans. AIME 1943, 151, 57-72. (3) Fetkovich, M. J. A Simplified Approach to Water Influx CalculationssFinite Aquifer Systems. J. Pet. Technol. 1971, 814-828. (4) van Everdingen, A. F.; Hurst, W. The Application of the Laplace Transformation to Flow Problems in Reservoirs. Trans. AIME 1949, 186, 305-324. (5) Coats, K. H. A Mathematical Model for Water Movement About Bottom-Water Drive Reservoirs. SPE J. 1962, 225, 44-52. (6) Allard, D. R.; Chen, S. M. Calculation of Water Influx for BottomWater Drive Reservoirs. SPE Res. Eng. 1988, 369-379. (7) Fanchi, J. R. Analytical Representation of the van EverdingenHurst Influence Functions for Reservoir Simulation. SPE J. 1985, 405-406. (8) Carter, R. D.; Tracy, G. W. An Improved Method for Calculating Water Influx. J. Pet. Technol. 1960, 58-60. (9) Klins, M. A.; Bouchard, A. J.; Cable, C. L. A Polynomial Approach to the van Everdingen-Hurst Dimensionless Variables for Water Encroachment. SPE Res. Eng. 1988, 320-326. (10) Hurst, W. Simplification of the Material Balance Formulas by the Laplace Transformation. Trans. AIME 1958, 213, 292-308. (11) Leung, W. F. A Fast Convolution Method for Implementing Single-Porosity Finite/Infinite Aquifer Models for Water-Influx Calculations. SPE Res. Eng. 1986, 490-510. (12) Leung, W. F. A New Pseudosteady-State Model for DualPorosity/Dual-Permeability Aquifers and Two Interconnected SinglePorosity Aquifers. SPE Res. Eng. 1986, 511-520. (13) Vogt, J. P.; Wang, B. Accurate Formulas for Calculating the Water Influx Superposition Integral. Proceedings of the SPE Eastern Regional Meeting, Pittsburgh, PA, 1987; pp 21-23.

10.1021/ef980128q CCC: $18.00 © 1999 American Chemical Society Published on Web 12/03/1998

Neural Network Model

Energy & Fuels, Vol. 13, No. 1, 1999 89

practically all flow regimes, provided that the flow geometry is actually radial. The solution of van Everdingen and Hurst is for both the constant-terminal-rate case and the constant-terminal-pressure case of finite and infinite edge-water aquifers only, however, it does not cover the bottom-water drive reservoirs. Coats5 developed a model that takes into consideration the vertical flow of water into the reservoir. However, his model has two major setbacks: (1) the presented solution is for the constant-terminal-rate case only, which permits the calculation of the pressure from a known water influx rather than the reverse, and (2) the model is applicable only to infinite aquifers and does not provide a solution for the finite aquifers. Allard and Chen6 expanded Coats5 model for bottom-water drive reservoirs by presenting a constant-terminal-pressure solution that covers finite and infinite aquifers. Even though the van Everdingen-Hurst,4 Coats,5 and Allard-Chen6 models offer accurate solutions for edgeand bottom-water drive reservoirs, they suffer a major limitation because the results of these models are presented in table forms, which greatly limits their application in computer analysis and reservoir simulation studies. Fanchi7 developed analytical expressions for the constant-terminal-rate solution of the van EverdingenHurst4 model. However, this model does not yield values for all aquifer sizes. Fanchi’s7 expressions are best suited for the Carter-Tracy8 method. On the other hand, Klins et al.9 presented a polynomial approach to the van Everdingen-Hurst4 model that covers both the constant-terminal-rate and the constant-terminal-pressure solutions. Even though Klins et al.’s9 model is general, its application is not straightforward and involves many complexities and much computational effort. No simple model has been developed to determine the dimensionless water influx or the dimensionless pressure for bottom-water drive reservoirs. Computer simulation or table lookup and interpolation between time entries and tables are still used for this purpose. These types of calculations introduce a major difficulty and time delay since water-influx values are involved in most reservoir engineering studies on a daily basis. It is not surprising, therefore, that there is an urgent need for simple equations and models that are capable of providing accurate and fast access for this type of information. The objective of this paper is to present neural network models that are suitable for edge- and bottomwater drive reservoirs. The models will provide solutions for both the constant-terminal-rate case and the constantterminal-pressure case of finite and infinite aquifers with a high degree of accuracy and minimum time requirements.

Figure 1. Ideal radial flow model for edge-water drive system.

One of the major assumptions used in the development of eq 1 is that the flow in the reservoir is radial, as illustrated in Figure 1. Equation 1 can be expressed in dimensionless form as

δ2pD δrD2

+

1 δpD δpD ) rD δrD δtD

(2)

where the dimensionless time tD, dimensionless radius rD, and dimensionless pressure pD are, respectively, defined as

tD )

0.0002637kt φµcrR2

(3)

r rR

(4)

pi - p pi - pwf

(5)

rD ) pD )

van Everdingen and Hurst4 developed two solutions for eq 1: (1) the constant-terminal-rate solution where the flow rate across the boundary is considered constant and the pressure drop throughout the reservoir is calculated as a function of time and (2) the constantterminal-pressure solution where the pressure at the boundary is assumed constant and the flow rate is determined as a function of time. For any time-dependent pressure drop, the water influx into the reservoir can be calculated using the principle of superposition as i-1

We(tDi) ) B

(∆pj)WeD(tDi - tDj) ∑ j)0

(6)

where 2. Process Modeling 2.1. Edge-Water Drive Reservoirs. The unsteadystate flow of a slightly compressible fluid in a porous media is best described by the diffusivity equation as

δp δ2p 1 δp φµc ) + δr2 r δr 0.0002637k δt

(1)

∆pj )

pj-1 - pj+1 2

(7)

B is called the water-influx constant and is defined as

θ B ) 1.119φchrR2 360

(8)

90 Energy & Fuels, Vol. 13, No. 1, 1999

Nashawi and Elkamel

added a new term to the diffusivity equation (eq 1) to account for pressure gradients in the vertical direction due to water flow. This results in a new form of eq 1

δ2p

φµc δ2p δp 1 δp + ) + F k δr2 r δr δz2 0.0002637k δt

(10)

where Fk is the ratio of vertical to horizontal permeability defined as

Fk )

k kv

(11)

Equation 10 can be expressed in dimensionless form as

δ2pD Figure 2. Ideal flow model for bottom-water drive system.

The application of eq 6 is well-illustrated in several reservoir engineering texts.14,15 Two major limitations are involved in eq 6: (1) the application of superposition is not a straightforward procedure and (2) the values of the dimensionless water influx WeD are listed in tables as a function of dimensionless time and dimensionless reservoir to aquifer radius r′D. Thus, table lookup and sometimes interpolation between tables are required to determine the appropriate value of WeD. Hurst10 and Carter and Tracy8 developed another application of the constant-terminal-rate solution. They suggested that water influx can be calculated additively by the following equation:

We(tDj) ) We(tDj-1) + B∆p - We(tDj-1)p′D(tDj)

[

pD(tDj) - tDj-1p′D(tDj)

]

(tDj - tDj-1) (9)

Equation 9 succeeded in eliminating the drawback involved in using superposition. However, it did not solve the problem of table lookup and data interpolation, but instead, it added another difficulty to the calculations, which consists of determining the dimensionless pressure pD and its derivative p′D with respect to dimensionless time. van Everdingen and Hurst4 solved the diffusivity equation for the constant-terminal-rate case that allows the calculation of pD. However, as in the case of constant-terminal-pressure solution, van Everdingen and Hurst4 reported the values of the dimensionless pressure in table form as a function of dimensionless time and dimensionless reservoir to aquifer radius. Furthermore, p′D has to be calculated either analytically or numerically. 2.2. Bottom-Water Drive Reservoirs. Equation 1 was developed for radial flow of a slightly compressible fluid. This equation should not be applied when a significant vertical fluid movement from the bottom of the reservoir exists. Figure 2 illustrates an ideal bottomwater drive flow model showing upward flow of water into the oil zone. Coats5 and later Allard and Chen6 (14) Dake, L. P. Fundamentals of Reservoir Engineering; Elsevier Scientific Publishing Co.: New York, 1987; pp 315-324 (15) Craft, B. C.; Hawkins, M. F. Jr; Terry, R. E. Applied Petroleum Reservoir Engineering, 2nd ed.; Prentice Hall Inc.: Englewood Cliffs, NJ, 1991; pp 280-300.

δrD2

+

2 1 δpD δ pD δpD + ) rD δrD δz 2 δtD

(12)

D

where zD is the dimensionless distance defined as

zD )

z rRFk1/2

(13)

Coats5 presented an analytical solution for eq 12 for the constant-terminal-rate case of an infinite aquifer wherein the rate of water encroachment across the reservoir aquifer interface is specified. The basic solution obtained by Coats5 is given by

p ) p0 -

282ewµ rRkFk1/2

pD(tD)

(14)

Equation 14 is employed to determine the reservoir pressure p0 as a function of time for a constant rate of water influx ew. The general solution for any timedependent rate can be obtained by applying the principle of superposition to eq 14

pj ) p0 -

282µ rRkFk1/2

i)j-1

∑ i)0

∆ewipDj-1

(15)

The values of dimensionless pressure pD are given by Coats5 in table form as a function of dimensionless time tD and dimensionless thickness constant z′D. tD was defined by eq 3, whereas z′D is given by

z′D )

h rRFk1/2

(16)

Thus, for a specific value of z′D, the dimensionless pressure pD is a function of dimensionless time tD only. More recently, Allard and Chen6 expanded Coats5 work by presenting a constant-terminal-pressure solution for both finite and infinite aquifer sizes. The solution of Allard and Chen6 is given by the following equation

We ) B′∆pWeD

(17)

Equation 17 allows the calculation of water influx into the reservoir as a function of a specific pressure drop. Applying the principle of superposition to eq 17 results



Table 1. Scaling Groups Characterizing Various Types of Reservoirs type of driving mechanism

scaling groups

functional form

Edge Water constant-terminalpressure solution finite aquifers infinite aquifers constant-terminalrate solution finite aquifers infinite aquifers

WeD, tD, r′D WeD, tD

WeD ) f(tD, r′D) WeD ) f(tD)

pD, tD, r′D pD, tD

pD ) f(tD, r′D) pD ) f(tD)

Bottom Water constant-terminalpressure solution finite aquifers infinite aquifers constant-terminalrate solution finite aquifers infinite aquifers

WeD, tD, r′D, z′D WeD, tD, z′D

WeD ) f(tD, r′D, z′D) WeD ) f(tD, z′D)

pD, tD, r′D, z′D pD, tD, z′D

pD ) f(tD, r′D, z′D) pD ) f(tD, z′D)

in a more general solution for any time-dependent pressure drop as i

Wei ) B′

∆pjWeD i-j+1 ∑ j)1

(18)

Equation 18 is similar to the one proposed by van Everdingen and Hurst4 (eq 6) with the exception that B′ is defined as

B′ ) 1.119φchrR2

Similarly, for the constant-terminal-rate solution, the dimensionless pressure is given by

pD ) f(tD, r′D, z′D)

(21)

(19)

Allard and Chen6 used a numerical simulator to calculate the values of dimensionless water influx. In a fashion similar to van Everdingen and Hurst4 and Coats,5 they presented their solution in table form, however, due to the presence of vertical water movement, their solution is a function of one additional dimensionless variable, the dimensionless thickness constant z′D defined by eq 16. Therefore, for a certain aquifer size, a fixed value of r′D, and a known value of z′D, the dimensionless water influx WeD is a function of the dimensionless time tD only. The calculation of the cumulative water influx using eq 18 is well-explained in the literature.14,15 3. Artificial Neural Network To prepare artificial neural network models for prediction of the dimensionless water influx and the dimensionless pressure, the input parameters characterizing the flow must be decided upon. From the previous section, the scaling groups characterizing the flow depend on the type of reservoir, edge- or bottom-water drive, and the aquifer size, finite or infinite. These groups are summarized in Table 1. The groups completely characterize the flow phenomena for each situation encountered. For instance, for the constantterminal-pressure solution, finite bottom-water drive reservoir (row 5), the dimensionless water influx can be expressed in functional form as

WeD ) f(tD, r′D, z′D)

Figure 3. Feedforward neural network architecture used for predicting water influx for a finite bottom-water drive reservoir.

(20)

The inputs to the artificial neural network modeling this situation are then tD, r′D, and z′D, and the output is WeD.

Hence, a neural network for the constant-terminal-rate solution would have three inputs tD, r′D, and z′D and one output pD. A neural network approach for modeling water influx differs from the traditional numerical approaches discussed earlier in several important ways. A neural network does not require information or assumptions about the partial differential equations modeling the process nor does it require parameter values such as physical and petrophysical properties characterizing the process. This is advantageous for the process under consideration where it is difficult to obtain parameter values and the process itself is not well understood. A neural network, on the other hand, uses input-output parameters to be trained to recognize the correct relationship. Once successfully trained, the network can be employed to compute the values of the output variable for inputs that are similar but not necessarily the same as those used in training. The training data may be assembled from experimental data, past field data, numerical reservoir simulation, or a combination of these. Artificial neural networks are computing systems made up of simple interconnected processing elements called neurons.16,17 The neurons are arranged in layers. The network for the prediction of dimensionless water influx for a constant-terminal-pressure solution, finite bottom-water drive reservoir is shown in Figure 3. The input layer of the network receives information from the (16) Caudill, M.; Butler, G.Understanding Neural Networks, Computer Explorations; The MIT Press: Cambridge, MA 1982; Vols. 1 and 2. (17) Caudill, M. Neural Networks Primer; Miller Freeman Publications: San Francisco, CA, 1989.


Nashawi and Elkamel

Table 2. Number of Training and Testing Data Points and Neurons Used in the Preparation of the Various NN Models type of driving mechanism

no. of data points in the training data set

no. of data points in the testing data set

no. of neurons in the hidden layer

465 493

51 54

7 4

278 42

30 5

6 5

1,100 3,316

113 342

15 15

461

47

10

Edge Water constant-terminal-pressure solution finite aquifers infinite aquifers constant-terminal-rate solution finite aquifers infinite aquifers

Bottom Water constant-terminal-pressure solution finite aquifers infinite aquifers constant-terminal-rate solution infinite aquifers

outside world, processes it, and then transmits it to the hidden layers. The output layer communicates the prediction of dimensionless water influx to the outside world. The neurons are connected by weights. These weights represent the fitting parameters of the network. As can be seen in Figure 3, there are massive connections among the neurons, thus, giving the network an extensive ability to correlate input-output information. The central element of an artificial neural network is the neuron or the processing element. A neuron processes an input vector I with components, I1, I2, I3, ..., In to give an output a. This output can serve as an input to other neurons. Several factors other than the input vector I determine the output a. One such factor is the weight vector w that multiplies the input vector I in order to obtain a weighted input on which further calculations are performed. Another influencing factor is the bias b of the neuron. This bias is subtracted from the weighted sum, giving rise to a total activation (TA) of the neuron (TA ) ΣWiIi - b). The output of the neuron is obtained by applying a transfer function f on the total activation (i.e., a ) f(TA)). There are several forms of transfer functions that can be applied. In this study, a sigmoid function is used for the input and hidden neurons and a linear function is applied for the output neurons. Therefore, the output of a neuron with a sigmoidal transfer function is given by

a)

1 1 + exp(-TA)

(22)

where TA is the activation of the neuron. For a hidden neuron, the activation is calculated using the input to the neuron, which is

Ii )

∑j (wjiaj)

(23)

where the sum over j represents all the neurons in the preceding layer. The weights wji are associated with the connections from the j neuron to the i neuron. The calculated outputs from the hidden neurons are summedup in a similar fashion to form the input to neurons in subsequent layers after being modified by the appropriate connection weights. There are various methods to determine the connection weights in a neural network. In this study, the back-propagation algorithm is employed. The weights are first randomly initiated. The network is then subjected to the input-output data with the objective

of finding the best weights that minimize the sum of squared error between the actual values and the values predicted by the neural network. To speed-up the convergence properties of the back-propagation algorithm, the Levenberg-Marquardt algorithm for optimizing a nonlinear function is applied with a momentum term.18 A variable learning rate is employed in order to make sure that the training step size is as large as possible but still stays within the stability limit of the algorithm. The training data used to prepare the artificial neural networks for predicting the dimensionless water influx and the dimensionless pressure were obtained from computer simulation runs and from the literature. The data are divided into different groups, as shown in Table 1. This division is necessary since the scaling groups characterizing a given reservoir situation are different from one case to another. More discussion on this will be given in the Results section. 4. Results As mentioned earlier, several neural networks were prepared to simulate aquifer systems. For each system, the collected data were divided into two sets: (1) a training set and (2) a testing and cross-validating set. The testing set is about 10% of the overall data set for each case and was randomly selected using a generator based on a formula that mathematically computes random numbers given an initial feed.19 The number of data points within each set considered is given in Table 2. The finite bottom-water drive constant-terminalrate solution was not considered due to the lack of simulation models in the literature. Since the input and output variables to the neural networks are not of the same order of magnitude, the training algorithm (back-propagation) has to compensate by adjusting the network weights. For instance, if input variable 1 (e.g., tD) has a value of 500 and input variable 2 (e.g., r′D) has a value of 10, the assigned weight for the second variable entering a node of the hidden layer must be much greater than that of the first variable in order for variable 2 to have any significance. This is not very effective during the training process. In addition, the sigmoidal transfer function is insensi(18) Reklaitis, G. V.; Ravindra, A.; Ragsdell, K. M. Engineering Optimization; John Wiley and Sons: New York, 1983. (19) Press, W.; Flannery, B.; Teukolsky, S.; Vetterling, W. Numerical Recipes: the Art of Scientific Computing; Cambridge University Press: London, 1986.



Table 3. Range of the Input and the Output Variables Used in Training the Various NN Models dimensionless time

type of driving mechanism

min

max

dimensionless water influx/pressure

dimensionless radius

min

max

min

max

49.36 1.42E+11

1.5

10

1.5

10

4

10

dimensionless thickness constant min

max

0.05 0.05

1.0 1.0

0.05

1.0


0.05 0.01

500 2E+12

0.276 0.112

0.06 0.01

70 1000

0.251 0.112

3.008 3.860

Bottom Water constant-terminal-pressure solution finite aquifers infinite aquifers constant-terminal-rate solution infinite aquifers

2.0 0.1

1200 2E+12

1.507 0.176

49.96 1.40E+11

0.1

1600

0.285

43.45

Table 4. Statistical Results type of driving mechanism

min absolute relative error (%)

max absolute relative error (%)

av absolute relative error (%)

0.00043 0.00093

0.64110 0.71672

0.12351 0.14556

0.00002 0.00269

0.83821 0.95397

0.03905 0.06038

0.00002 0.00044

0.30895 0.10272

0.04229 0.03122

0.00000 0.00018

0.30311 0.20535

0.04596 0.12636

0.00009 0.00249

0.92396 0.65045

0.16210 0.17164

0.00003 0.00006

1.63043 1.10279

0.07889 0.07399

0.00001 0.00005

0.32901 0.34325

0.02457 0.02598

Edge Water constant-terminal-pressure solution finite aquifers training data testing data infinite aquifers training data testing data constant-terminal-rate solution finite aquifers training data testing data infinite aquifers training data testing data

Bottom Water constant-terminal-pressure solution finite aquifers training data testing data infinite aquifers training data testing data constant-terminal-rate solution infinite aquifers training data testing data

tive to large values (e.g., f(5) ) 0.993; f(50) ) 1; and f(500) ) 1). To avoid the above problems, all the input and output variables have been scaled before training. The following scaling rule for a given input or output variable X has been applied

Xscaled )

X - Xmin Xmax - Xmin

(24)

where Xmin and Xmax are the minimum and maximum values, respectively, of X over the range of the data. The range of the input and the output variables for all cases considered in this research is presented in Table 3. In addition, to speed-up training, it was found necessary, except for the finite edge-water constantterminal-pressure solution, to further normalize the scaled variables by taking their natural logarithm. To determine a neural network with a good predictive ability, various architectures were tried. For each architecture with a certain number of hidden neurons, the best weights that minimize the sum of squared error were obtained using the training data set. The predic-

Figure 4. Training data pointssedge-water drive reservoirs constant-terminal-pressure solution (finite aquifers).

tions of the network are then checked against the testing data set. The statistical results including the minimum absolute relative error, the maximum abso-


Nashawi and Elkamel

Figure 5. Training data pointssedge-water drive reservoirs constant-terminal-pressure solution (infinite aquifers).

Figure 8. Training data pointssbottom-water drive reservoirs constant-terminal-pressure solution (finite aquifers).

Figure 6. Training data pointssedge-water drive reservoirs constant-terminal-rate solution (finite aquifers).

Figure 9. Training data pointssbottom-water drive reservoirs constant-terminal-pressure solution (infinite aquifers).

Figure 7. Training data pointssedge-water drive reservoirs constant-terminal-rate solution (infinite aquifers).

Figure 10. Training data pointssbottom-water drive reservoirs constant-terminal-rate solution (infinite aquifers).

lute relative error, and the average absolute relative error are shown in Table 4. For all cases one hidden layer was found to give good predictions in terms of both the training and testing data sets. The number of neurons of the hidden layer is given in Table 2. Figures 4-10 illustrate the percent error between the predicted and actual dimensionless water-influx/pressure for the

training data sets. On the other hand, Figures 11-16 display the percent error of the testing data sets. As can be seen, the maximum percent error for all cases is less than 1% except for the constant-terminal-pressure solution of infinite bottom-water drive reservoirs where it is 1.63%. This indicates that the prepared neural network models will yield accurate predictions in simu-


Figure 11. Testing data pointssedge-water drive reservoirs constant-terminal-pressure solution (finite aquifers).

Figure 12. Testing data pointssedge-water drive reservoirs constant-terminal-pressure solution (infinite aquifers).

Figure 13. Testing data pointssedge-water drive reservoirs constant-terminal-rate solution (finite aquifers).

lating water influx. In addition to their accuracy, the prepared neural networks have the added advantage of furnishing predictions quickly. Unlike most reservoir simulators that require a long computing time and large storage capacity, a neural-network-based simulator can furnish accurate results in seconds. Furthermore, the NN models represent an excellent substitute for the tabular listings of van Everdingen and Hurst,4 Coats,5 and Allard and Chen.6 Since the input variables tD, r′D,


Figure 14. Testing data pointssbottom-water drive reservoirs constant-terminal-pressure solution (finite aquifers).

Figure 15. Testing data pointssbottom-water drive reservoirs constant-terminal-pressure solution (infinite aquifers).

Figure 16. Testing data pointssbottom-water drive reservoirs constant-terminal-rate solution (infinite aquifers).

and z′D are implicit in the calculations, no interpolation is required. Furthermore, the traditional tabular listings cannot be easily integrated within general reservoir management programs to determine the aquifer effect on production and require large computer memory. On the other hand, the results from the NN models can be generated easily from the network weights and biases. These are given in Tables 5-11 for the various aquifer cases considered.


Nashawi and Elkamel

Table 5. Connection Weights and Biases in the ANN Modeling Edge-Water Drive ReservoirssConstant-Terminal-Pressure Solution (Finite Aquifers) i

W1i

W2i

bi

Woi

bo

1 2 3 4 5 6 7

16.3166913 13.8179362 -9.3669622 -8.2634214 0.2221339 -8.2229245 -9.3714239

4.2555194 15.3419878 3.8727589 6.6540290 0.0339562 6.7655817 24.9265204

-5.0200845 -30.8179834 2.8642130 1.9643394 -1.3025841 1.8266020 2.3702179

0.0064490 -0.0160591 -0.2714994 6.0933778 21.7322042 -5.5908488 0.0647820

-4.9751008

Table 6. Connection Weights and Biases in the ANN Modeling Edge-Water Drive ReservoirssConstant-Terminal-Pressure Solution (Infinite Aquifers) i

W1i

bi

Woi

bo

1 2 3 4

-28.1484834 9.1105957 32.8873740 0.5775264

5.3399271 -0.2198457 -2.9128030 -0.6940498

0.0035194 -0.1474181 -0.0062900 7.9211326

-2.5763191

Table 7. Connection Weights and Biases in the ANN Modeling Edge-Water Drive ReservoirssConstant-Terminal-Rate Solution (Finite Aquifers) i

W1i

W2i

bi

Woi

bo

1 2 3 4 5 6

1.9905954 -1.7438808 3.3899547 -2.2381447 2.8973439 7.2444143

-2.5823743 2.5011143 -0.0494594 2.6681829 0.7491556 27.2268028

-0.6218659 0.4170809 0.6259983 0.8200664 0.5306543 -3.4764226

-753.5467656 -358.8764036 24.0696173 -394.0033791 -19.8678682 -0.0487104

749.9473893

Table 8. Connection Weights and Biases in the ANN Modeling Edge-Water Drive ReservoirssConstant-Terminal-Rate Solution (Infinite Aquifers) i

W1i

bi

Woi

bo

1 2 3 4 5

1133.2647027 1.5462361 218.6375657 17.2778341 9.3503294

-295.4533539 1.8570115 -42.9188558 -6.0560522 -4.6857710

0.0056711 8.5636545 0.0154526 0.0368536 0.0624914

-7.4077749

Table 9. Connection Weights and Biases in the ANN Modeling Bottom-Water Drive ReservoirssConstant-Terminal-Pressure Solution (Finite Aquifers) i

W1i

W2i

W3i

bi

Woi

bo

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-10.5315522 1.1666981 27.0400736 -2.4422429 2.0785935 -1.6622105 -0.2685868 -1.7846807 -77.1603963 2.2195374 -1.2680070 0.8372340 -1.0267469 -0.4037190 0.6306157

-0.3486753 -4.5398737 -122.1293090 -0.3888239 -0.5015891 -0.7034123 0.0211838 1.6604257 -123.2386383 -2.3042991 2.7656559 0.2453579 0.1322278 -0.0514909 -0.5288117

65.3086504 -2.1068554 306.7409935 -1.7182566 -4.6299655 3.6368580 0.1452483 7.2524463 -103.7503283 48.4023493 2.7918415 -2.4448874 2.4114256 1.4246613 8.2216658

-54.8039330 2.6128439 -702.9224299 9.2762493 -1.8347768 -0.5978968 1.3905417 -10.6086358 -64.5706381 -76.1174689 -0.8765599 3.1775982 2.0543746 -1.6641080 -12.6312664

0.0135032 0.6019562 -0.0028897 -0.0345262 -0.3333457 0.1561230 -14.4051967 0.1691673 0.0095319 -0.0398991 -0.2342617 1.4947654 2.8327580 3.4357185 0.1605160

6.1768963

One of the major objectives of this research is to present NN models that provide accurate results with a minimum number of computing coefficients, weights, and biases. A one hidden layer neural network with a sigmoidal transfer function for the input and hidden layers and a linear transfer function for the output layer was found to achieve this goal for all the investigated cases. To further test the prepared neural network models, their predictions were compared to the results obtained from the Klins et al.9 and Fanchi7 equations. Table 12 shows a statistical comparison of the simulated data versus the predicted values obtained from the different methods. This comparison illustrates the many advan-

tages that the NN models have over the other models. A few of these advantages are (1) the neural networks provide values for the various bottom-water drive solutions and aquifer sizes, whereas the Klins et al.9 and Fanchi7 equations do not, (2) Fanchi7 expressions are only applicable for the constant-terminal-rate solution of edge-water drive reservoirs, while the NN models cover both the constant-terminal-rate and the constantterminal-pressure solutions, (3) furthermore, Fanchi7 expressions do not provide values for all the aquifer sizes presented by van Everdingen and Hurst,4 whereas the NN models do, thus we had, for instance, to derive equations similar to those of Fanchi7 for the missing aquifer sizes before we could perform the comparison,



Table 10. Connection Weights and Biases in the ANN Modeling Bottom-Water Drive ReservoirssConstant-Terminal-Pressure Solution (Infinite Aquifers) i

W1i

W2i

bi

Woi

bo

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

-26.9636406 -30.3812356 47.3735106 46.5453142 57.3036961 -30.2985313 26.1352429 94.0772906 -3.4613925 -5.7274405 -1.5783779 45.3283352 -10.3571886 -45.9420561 -34.4600062

-7.0019635 1.0159611 19.1238577 31.4896229 25.4916000 1.0117912 -5.4734347 12.6947650 0.0007764 0.0005779 6.8189163 30.0072275 0.0367131 -30.7532683 2.1878727

8.2223354 6.1722489 -16.3582484 -11.1642081 -32.9610335 6.1532177 4.8405916 -12.7823734 3.7316292 3.0215499 -15.0944879 -10.8273577 3.1498371 10.9976863 2.5725734

-0.0056536 2.4764736 -0.0016365 6.8848348 0.0002134 -2.5050995 0.0726494 0.0036481 -1.1261705 -0.3409044 -20.4450188 6.8481096 -0.1287196 13.7286748 -0.0241218

-12.1477487

Table 11. Connection Weights and Biases in the ANN Modeling Bottom-Water Drive ReservoirssConstant-Terminal-Rate Solution (Infinite Aquifers) i

W1i

W2i

bi

Woi

bo

1 2 3 4 5 6 7 8 9 10

-10.0123414 -4.3681513 0.5431513 1.5821213 9.6394388 -0.0301060 -0.0980442 6.8052418 -10.2630255 -0.5842752

7.8504439 -0.0235151 -67.4934758 -3.5021745 -8.3857515 -28.8115498 20.3766562 -0.0922827 7.2577838 1.6966112

-8.0192026 -1.3245817 63.6363193 1.7723716 8.4793010 21.7556366 -5.2344649 -4.7022364 -7.4760587 0.7685161

-15.2528078 -2.3452726 0.0108361 0.1954750 -6.4947765 0.0323358 -0.0599862 0.0164555 9.1464174 -1.1575765

7.8881910

Table 12. Comparison of Absolute Relative Errors of Different Models Klins et al.9

NN model type of driving mechanism

Fanchi7

min max av min max av min error (%) error (%) error (%) error (%) error (%) error (%) error (%)

max error (%)

av error (%)


0.00043 0.00002

0.71672 0.95397

0.12569 0.04116

0.00000 0.00044

122.82634 2.02162 1.32629 0.27093

no model is provided by Fanchi7 for the constant-terminalpressure solution

0.00002 0.00000

0.30895 0.30311

0.04121 0.05452

0.00005 0.00031

2.76358 0.72786

0.00004 0.04020

0.05631 0.11250

0.50002 9.94968

0.04592 1.40355

Bottom Water constant-terminal-pressure solution finite aquifers infinite aquifers constant-terminalrate solution infinite aquifers

0.00009 0.00003

0.92396 1.63043

0.16299 0.07843

0.00001

0.34325

0.02470

no models are provided by either Klins et al.9 or Fanchi7

(4) the results obtained from the NN models are more accurate than those obtained using Klins et al.9 and Fanchi7 methods, and (5) the NN models are superior to Klins et al.9 and Fanchi7 equations in the sense that they provide accurate results with minimum time requirement and less computational complexity. The developed neural network model can also be used through a parallel semi-parametric technique that combines the network output with estimates of the previous correlations. This hybrid approach can be applied, for instance, to impose the smoothness of the dimensionless water influx function or to restrict this function to certain bounds. The merits of such an approach are discussed in detail by Psichogios and Ungar20 and Thompson and Kramer.21 In this study, the predicted dimensionless variables were all within rea-

sonable ranges and the straightforward training method that has been employed proved to yield good results for most field applications. In addition, the training data used cover all possible ranges and there is no need to direct the neural network with previous correlations in regions of sparse data (Table 3). Furthermore, some of the previous correlations themselves, as previously mentioned, do not even compass all possible reservoir conditions, and in this respect, the neural network is highly superior to them and cannot be supplemented with them (Table 12).

(20) Psichogios, D. C.; Ungar, L. H. A Hybrid Neural Network-First Principles Approach to Process Modeling. AIChE J. 1992, 38 (10), 1499-1511.

(21) Thompson, M. L.; Kramer, M. A. Modeling Chemical Processes Using Prior Knowledge and Neural Networks. AIChE J. 1994, 4 (8), 1328-1340.

5. Conclusion NN models that can predict the dimensionless variables for edge- and bottom-water drive reservoirs have


been presented. The values obtained from the models are as accurate as the original values reported in table forms by van Everdingen and Hurst,4 Coats,5 and Allard and Chen.6 The models use less computational time than the traditional table look up. Because the characterizing inputs are implicit in the calculations, the desired dimensionless output variables can be obtained without interpolation. When compared to Klins et al.9 and Fanchi7 methods, the NN models yield more accurate results with less computational time and complexity. Acknowledgment. We express our appreciation to the Kuwait University Research Administration for financially supporting this work through a University Research Grant (EP-017). Nomenclature a ) output of a neuron calculated from a total activation and a transfer function b ) bias of a neuron B ) water-influx constant for edge-water drive reservoirs, eq 8, bbl/psi [m3/kPa] B′ ) water-influx constant for bottom-water drive reservoirs, eq 19, bbl/psi [m3/kPa] ew ) water-influx rate, bbl/day [m3/day] c ) effective aquifer compressibility, psi-1 [kPa-1] f ) transfer function

Nashawi and Elkamel Fk ) ratio of vertical to horizontal permeability, dimensionless h ) aquifer thickness, ft [m] I ) input vector to the neural network k ) horizontal permeability, md kv ) vertical permeability, md p ) pressure at the oil-water contact, psia [kPa] p0 ) initial aquifer and reservoir pressure at reservoir depth, psia [kPa] pD ) dimensionless pressure p′D ) dimensionless pressure derivative ∆p ) pressure drop at the oil-water contact, psia [kPa] r ) radius, ft [m] rD ) dimensionless radius r′D ) dimensionless radius, ratio of aquifer to reservoir size re ) aquifer radius, ft [m] rR ) reservoir radius, ft [m] t ) time, h tD ) dimensionless time TA ) total activation of a neuron wji ) weights in the neural network model We ) cumulative water influx, bbl [m3] WeD ) dimensionless water influx z ) vertical distance coordinate, ft [m] zD ) dimensionless vertical distance coordinate z′D ) dimensionless thickness constant θ ) angle subtended by the reservoir circumference µ ) formation water viscosity, cp [mPa s] φ ) formation porosity, fraction EF980128Q

Neural Network Model for the Prediction of Water Aquifer

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